SAT Math Algebra practice: 9 worksheets with answer keys

Linear equations in one variable

Linear equations in one variable · 24 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. $x + 16 = 30$. What value of $x$ is the solution to the given equation? (Grid-in)
2. If $6n = 12$, what is the value of $n + 4$?
   • A) 2
   • B) 6
   • C) 8
   • D) 12
3. Three more than eight times a number $x$ is equal to 83. Which equation represents this situation?
   • A) $3x + 8 = 83$
   • B) $8x + 3 = 83$
   • C) $3(x + 8) = 83$
   • D) $8(x + 3) = 83$
4. $x - 53 = \frac{91}{2}$. Which equation has the same solution as the given equation?
   • A) $2x - 53 = 91$
   • B) $2x - 106 = 91$
   • C) $x = \frac{38}{2}$
   • D) $x = \frac{144}{2}$
5. $7x(2 - 3) = 63$. Which equation has the same solution as the given equation?
   • A) $-7x = 63$
   • B) $7x - 3 = 63$
   • C) $14x - 3 = 63$
   • D) $-1x = 63$
6. If $\frac{k}{5} = 15$, what is the value of $k - 10$?
   • A) 5
   • B) 10
   • C) 65
   • D) 75

Level 2: Medium (Standard SAT)

1. If $4x - 28 = -24$, what is the value of $x - 7$?
   • A) $-24$
   • B) $-6$
   • C) $-1$
   • D) $1$
2. $14x = 2w + 19y$. The given equation relates the distinct positive real numbers $w$, $x$, and $y$. Which equation correctly expresses $w$ in terms of $x$ and $y$?
   • A) $w = 7x - \frac{19}{2}y$
   • B) $w = 14x - 19y$
   • C) $w = \frac{14x + 19y}{2}$
   • D) $w = 7x - 19y$
3. The equation $d = 30t$ gives the distance $d$, in feet, that a bus will travel $t$ seconds after passing a marker. How many feet from the marker will the bus be 2.5 seconds after passing it? (Grid-in)
4. A veterinarian recommends that each day a certain rabbit should eat 25 calories per pound of the rabbit's weight, plus an additional 11 calories. Which equation represents this situation, where $c$ is the total number of calories if the rabbit's weight is $x$ pounds?
   • A) $c = 25x$
   • B) $c = 11x + 25$
   • C) $c = 25x + 11$
   • D) $c = 36x$
5. $5.25x + 7.50y = 500$. The equation represents the total sales in dollars from selling two different-sized containers of fruit, where $x$ is the number of smaller containers sold and $y$ is the number of larger containers sold. According to the equation, what is the price, in dollars, of each smaller container? (Grid-in)
6. The equation $V = P(1 + rt)$ gives the total value $V$ of an investment of $P$ dollars at a simple annual interest rate $r$ for $t$ years. Which equation correctly expresses $r$ in terms of $V$, $P$, and $t$?
   • A) $r = \frac{V - 1}{Pt}$
   • B) $r = \frac{V - P}{P}$
   • C) $r = \frac{V - P}{Pt}$
   • D) $r = \frac{V}{Pt} - P$

Level 3: Hard (Advanced SAT)

1. $8x + 12 = kx + 12$. In the given equation, $k$ is a constant. If the equation has infinitely many solutions, what is the value of $k$? (Grid-in)
2. $|2x - 5| = 11$. What is the positive solution to the given equation? (Grid-in)
3. $cx - 64 = 48$. In the given equation, $c$ is a constant. If $x = 4$ is a solution to the equation, what is the value of $c$?
   • A) 16
   • B) 28
   • C) 64
   • D) 112
4. The formula $A = \frac{1}{2}bh$ gives the area $A$ of a triangle with base $b$ and height $h$. Which equation correctly expresses $h$ in terms of $A$ and $b$?
   • A) $h = \frac{A}{2b}$
   • B) $h = \frac{2A}{b}$
   • C) $h = 2Ab$
   • D) $h = A - \frac{1}{2}b$
5. $5(x - 3) + 2x = px - 15$. In the given equation, $p$ is a constant. If the equation has no solution, what is the value of $p$?
   • A) 2
   • B) 3
   • C) 5
   • D) 7
6. $2(x - 3) + kx = 3x - 5$. In the given equation, $k$ is a constant. If the equation has no solution, what is the value of $k$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. $3x + 15 = 3(x + p)$. In the given equation, $p$ is a constant. If the equation has infinitely many solutions, what is the value of $p$?
   • A) 3
   • B) 5
   • C) 15
   • D) It cannot be determined.
2. $2(jx + 5) = 4x + k$. In the given equation, $j$ and $k$ are constants. If the equation has no solution, which of the following statements must be true?
   • A) $j = 2$ and $k \neq 10$
   • B) $j = 2$ and $k = 10$
   • C) $j \neq 2$ and $k = 10$
   • D) $j \neq 2$ and $k \neq 10$
3. If $\frac{x}{y} = 4$ and $\frac{x}{ny} = 4$, what is the value of $n$? (Grid-in)
4. $|x - 5| = c$. In the given equation, $c$ is a constant. If the equation has exactly one solution, what is the value of $c$? (Grid-in)
5. $ax + b = cx + d$. In the given equation, $a$, $b$, $c$, and $d$ are constants. If $a = c$ and $b \neq d$, how many solutions does the equation have?
   • A) Zero
   • B) Exactly one
   • C) Exactly two
   • D) Infinitely many
6. $(m+1)x + 2n = 3x - 6$. In the given equation, $m$ and $n$ are constants. If the equation has infinitely many solutions, what is the value of $m+n$? (Grid-in)

Linear equations and inequalities in context

Linear equations in one variable · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. A plumber charges a flat call-out fee of $\$50$ and an additional $\$75$ for each hour worked. Which equation represents the total cost $C$, in dollars, for $h$ hours of work?
   • A) $C = 50h + 75$
   • B) $C = 75h + 50$
   • C) $C = 125h$
   • D) $C = 75h - 50$
2. A baker has 120 cups of flour. Every batch of cookies requires 3 cups of flour. Which equation models the number of cups of flour remaining, $F$, after baking $b$ batches of cookies?
   • A) $F = 120 + 3b$
   • B) $F = 3b - 120$
   • C) $F = 120 - 3b$
   • D) $F = 120b - 3$
3. The function $h(t) = 15 + 2.5t$ represents the height $h$, in inches, of a plant $t$ weeks after it was purchased. What is the best interpretation of the number 15 in this context?
   • A) The number of weeks the plant will survive.
   • B) The height of the plant when it was purchased.
   • C) The amount the plant grows each week.
   • D) The maximum height the plant can reach.
4. A theater can hold a maximum of 300 people. If $a$ represents the number of adults and $c$ represents the number of children attending a show, which inequality represents this constraint?
   • A) $a + c \le 300$
   • B) $a + c \ge 300$
   • C) $a - c \le 300$
   • D) $a + c = 300$
5. A gym membership costs $\$20$ per month plus a one-time joining fee of $\$45$. If Sarah has exactly $\$165$ to spend, for how many months can she afford the gym membership? (Grid-in)

Level 2: Medium (Standard SAT)

1. A catering company charges $\$15$ per guest for a standard meal and $\$25$ per guest for a premium meal. If a client has a budget of exactly $\$1,000$ and invites exactly 50 guests, which system of equations can be used to find the number of standard meals $s$ and premium meals $p$ ordered?
   • A) $s + p = 1000$ and $15s + 25p = 50$
   • B) $s + p = 50$ and $15s + 25p = 1000$
   • C) $15s + p = 50$ and $s + 25p = 1000$
   • D) $s - p = 50$ and $25s + 15p = 1000$
2. The equation $P = 100 + 20(w - 1)$ gives the profit $P$, in dollars, a salesperson makes after working for $w$ weeks, where $w \ge 1$. What is the salesperson's profit, in dollars, for the first week? (Grid-in)
3. An elevator has a maximum weight capacity of 1,500 pounds. Suppose the elevator currently contains 3 people whose average weight is 160 pounds. If a group of boxes weighing 40 pounds each is loaded into the elevator, what is the maximum number of boxes that can be safely loaded? (Grid-in)
4. A factory produces $m$ machines every hour. After 8 hours of production, 12 machines are rejected due to defects. If the factory produced 100 non-defective machines in those 8 hours, what is the value of $m$? (Grid-in)
5. The function $f(x) = 2.50x + 4.00$ defines the cost of a taxi ride where $x$ is the number of miles driven. If a passenger's total cost was $\$29.00$, how many miles did the taxi drive? (Grid-in)

Level 3: Hard (Advanced SAT)

1. A student is buying pens that cost $\$1.50$ each and notebooks that cost $\$3.00$ each. The student must buy at least 10 items in total but cannot spend more than $\$24.00$. If $p$ represents pens and $n$ represents notebooks, which system of inequalities represents this situation?
   • A) $p + n \ge 10$ and $1.50p + 3.00n \le 24$
   • B) $p + n \le 10$ and $1.50p + 3.00n \le 24$
   • C) $p + n \ge 10$ and $1.50p + 3.00n \ge 24$
   • D) $1.50p + 3.00n \ge 10$ and $p + n \le 24$
2. An internet provider charges a base fee of $\$40$ per month for up to 50 gigabytes (GB) of data. For each additional GB of data over 50, the company charges $\$2.50$. If a user's bill for a month was $\$75$, how many total GB of data did the user consume? (Grid-in)
3. At a high school, a teacher plans to buy laptops and tablets for the classroom. Laptops cost $\$400$ each and tablets cost $\$250$ each. The teacher must buy exactly 30 devices and has a budget of $\$9,000$. Assuming the teacher spends the entire budget, how many laptops were purchased? (Grid-in)
4. The equation $C = \frac{5}{9}(F - 32)$ converts a temperature in degrees Fahrenheit ($F$) to degrees Celsius ($C$). Which of the following equations correctly expresses $F$ in terms of $C$?
   • A) $F = \frac{9}{5}C + 32$
   • B) $F = \frac{9}{5}(C + 32)$
   • C) $F = \frac{5}{9}C - 32$
   • D) $F = \frac{9}{5}C - 32$
5. An artist is cutting wire to make sculptures. The wire is originally 120 inches long. The artist cuts $x$ pieces of wire that are 6 inches long and $y$ pieces of wire that are 8 inches long. If no wire is left over, which of the following could be the value of $x$?
   • A) 3
   • B) 5
   • C) 8
   • D) 11

Level 4: Very Hard (Slightly beyond SAT)

1. A farm produces both apples and peaches. On Monday, the farm sold $A$ pounds of apples for $\$1.20$ per pound and $P$ pounds of peaches for $\$1.50$ per pound. The total revenue for Monday was $\$360$. If the farm sold twice as many pounds of apples as peaches, what was the total number of pounds of fruit sold? (Grid-in)
2. The cost $C$, in dollars, to manufacture $n$ units of a certain product is given by $C = 1500 + 4.5n$. The manufacturer sells each unit for $\$12.00$. To make a profit of exactly $\$3,000$, how many units must the manufacturer produce and sell? (Assume Profit = Total Sales Revenue - Total Cost). (Grid-in)
3. In the equation $ax + by = c$, $a$, $b$, and $c$ are constants. The line represented by this equation in the $xy$-plane passes through the points $(0, 4)$ and $(6, 0)$. Which of the following could be the values of $a$, $b$, and $c$?
   • A) $a = 2, b = 3, c = 12$
   • B) $a = 4, b = 6, c = 24$
   • C) $a = 6, b = 4, c = 24$
   • D) $a = 3, b = 2, c = 12$
4. A train consists of passenger cars and cargo cars. An empty passenger car weighs 20 tons and an empty cargo car weighs 15 tons. The total weight of the empty train (excluding the locomotive) is exactly 400 tons. If there are 22 cars in total, how many more cargo cars are there than passenger cars? (Grid-in)
5. A shipping container holds $x$ small boxes and $y$ large boxes. The small boxes weigh 10 pounds each and have a volume of 2 cubic feet. The large boxes weigh 25 pounds each and have a volume of 4 cubic feet. The container is completely full with 400 cubic feet of boxes, and the total weight of the boxes is exactly 2,200 pounds. What is the value of $x$? (Grid-in)

Linear equations in two variables

Linear equations in two variables and their graphs · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. For the linear function $h$, $h(x) = x + b$, where $b$ is a constant. If $h(0) = 45$, what is the value of $b$? (Grid-in)
2. A line in the $xy$-plane has a slope of $\frac{1}{9}$ and passes through the point $(0, 14)$. Which equation represents this line?
   • A) $y = -\frac{1}{9}x - 14$
   • B) $y = -\frac{1}{9}x + 14$
   • C) $y = \frac{1}{9}x - 14$
   • D) $y = \frac{1}{9}x + 14$
3. Line $r$ in the $xy$-plane has a slope of $4$ and passes through the point $(0, 6)$. Which equation defines line $r$?
   • A) $y = 4x + 6$
   • B) $y = 4x - 6$
   • C) $y = 6x + 4$
   • D) $y = -6x + 4$
4. The function $f$ is defined by $f(x) = 2x + 3$. The graph of $y = f(x)$ in the $xy$-plane is parallel to line $j$. What is the slope of line $j$? (Grid-in)
5. The table below shows three values of $x$ and their corresponding values of $y$. There is a linear relationship between $x$ and $y$. {|c|c|} $x$ & $y$
   $0$ & $18$
   $1$ & $13$
   $2$ & $8$
   Which of the following equations represents this relationship?
   • A) $y = 18x + 13$
   • B) $y = -5x + 18$
   • C) $y = -5x + 13$
   • D) $y = 18x - 5$

Level 2: Medium (Standard SAT)

1. The table shows selected values from the linear function $f$. {|c|c|} $x$ & $f(x)$
   $0$ & $29$
   $1$ & $32$
   Which equation defines $f(x)$?
   • A) $f(x) = 29x + 32$
   • B) $f(x) = 32x + 29$
   • C) $f(x) = 3x + 29$
   • D) $f(x) = 3x + 32$
2. The relationship between two variables, $x$ and $y$, is linear. For every increase in the value of $x$ by 1, the value of $y$ increases by 8. When the value of $x$ is 2, the value of $y$ is 18. Which equation represents this relationship?
   • A) $y = 2x + 18$
   • B) $y = 8x + 2$
   • C) $y = 8x + 18$
   • D) $y = 2x + 8$
3. The function $f$ is defined by $f(x) = 7x - 84$. What is the $x$-intercept of the graph of $y = f(x)$ in the $xy$-plane?
   • A) $(-12, 0)$
   • B) $(0, -84)$
   • C) $(12, 0)$
   • D) $(84, 0)$
4. For the linear function $f$, $f(x) = 4x + b$, where $b$ is a constant. If $f(7) = 28$, what is the value of $b$? (Grid-in)
5. The table shows the linear relationship between the number of cars, $c$, on a commuter train and the maximum number of passengers and crew, $p$, that the train can carry. {|c|c|} Number of cars ($c$) & Maximum passengers and crew ($p$)
   $3$ & $174$
   $5$ & $284$
   $10$ & $559$
   Which equation represents the linear relationship between $c$ and $p$?
   • A) $p = 55c + 9$
   • B) $p = 55c - 9$
   • C) $c = 55p + 9$
   • D) $p = 110c + 9$

Level 3: Hard (Advanced SAT)

1. Line $k$ is defined by $y = 7x + \frac{1}{8}$. Line $j$ is perpendicular to line $k$ in the $xy$-plane. What is the slope of line $j$?
   • A) $-7$
   • B) $-\frac{1}{7}$
   • C) $\frac{1}{7}$
   • D) $7$
2. What is the equation of the line that passes through the points $(0, -8)$ and $(-8, 0)$ in the $xy$-plane?
   • A) $y = -x - 8$
   • B) $y = x - 8$
   • C) $y = -x + 8$
   • D) $y = x + 8$
3. A linear model predicts the number of active projects a company was working on $x$ months after the end of November 2012. If the model is defined by the equation $y = -x + 5$, what is the predicted number of active projects the company was working on at the exact end of November 2012? (Grid-in)
4. The linear function $f$ is defined by $f(x) = -2x + c$, where $c$ is a constant. The graph of $y = f(x)$ in the $xy$-plane intersects the $x$-axis at $(4, 0)$. What is the $y$-coordinate of the $y$-intercept of the graph of $f$? (Grid-in)
5. A line in the $xy$-plane passes through the points $(2, 5)$ and $(4, 9)$. What is the $y$-coordinate of the $y$-intercept of this line? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. Function $f$ is a linear function. The graph of $y = g(x)$ in the $xy$-plane is the result of shifting the graph of $y = f(x)$ down 3 units. If $g(x) = 4x - 5$, what is the $x$-coordinate of the $x$-intercept of the graph of $y = f(x)$? (Grid-in)
2. Line $p$ has the equation $ax + by = c$, where $a$, $b$, and $c$ are positive constants. What is the slope of a line that is perpendicular to line $p$ in the $xy$-plane?
   • A) $-\frac{a}{b}$
   • B) $\frac{a}{b}$
   • C) $-\frac{b}{a}$
   • D) $\frac{b}{a}$
3. For a linear function $h$, $h(3) = 14$ and $h(7) = 34$. What is the value of $h(10)$? (Grid-in)
4. The graph of linear function $f$ in the $xy$-plane passes through the origin and the point $(c, d)$, where $c \neq 0$. The graph of linear function $g$ passes through the origin and the point $(c, -d)$. If $f(5) = 10$, what is the value of $g(5)$? (Grid-in)
5. Line $L$ has the equation $y = 3x + b$, where $b$ is a constant. Line $M$ is perpendicular to line $L$ and passes through the point $(0, 4)$. If the two lines intersect at the point $(6, d)$, what is the value of $b$? (Grid-in)

Graphing linear relationships

Linear equations in two variables and their graphs · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. The equation of a line in the $xy$-plane is $y = -3x + 12$. What is the $x$-coordinate of the $x$-intercept of the line? (Grid-in)
2. A line in the $xy$-plane passes through the points $(0, 4)$ and $(2, 10)$. Which of the following is an equation of the line?
   • A) $y = 3x + 4$
   • B) $y = 4x + 2$
   • C) $y = 3x + 2$
   • D) $y = 2x + 4$
3. The graph of $4x + 5y = 40$ in the $xy$-plane has a $y$-intercept at $(0, b)$. What is the value of $b$? (Grid-in)
4. Line $p$ is defined by the equation $y = \frac{2}{3}x - 5$. Line $q$ is parallel to line $p$ and passes through the origin. What is the equation of line $q$?
   • A) $y = -\frac{3}{2}x$
   • B) $y = \frac{2}{3}x$
   • C) $y = \frac{2}{3}x + 5$
   • D) $y = -\frac{3}{2}x - 5$
5. A line has a slope of $-2$ and passes through the point $(4, 1)$. What is the $y$-intercept of this line? (Grid-in)

Level 2: Medium (Standard SAT)

1. Line $l$ is defined by the equation $2x - 7y = 14$. What is the slope of a line that is perpendicular to line $l$?
   • A) $-\frac{7}{2}$
   • B) $-\frac{2}{7}$
   • C) $\frac{2}{7}$
   • D) $\frac{7}{2}$
2. The function $f$ is defined by $f(x) = -\frac{1}{4}x + 6$. The graph of $y = f(x)$ is shifted up 3 units to create the graph of $y = g(x)$. What is the $x$-intercept of the graph of $g$? (Grid-in)
3. A line passes through the points $(-3, a)$ and $(5, a+16)$. What is the slope of the line?
   • A) $0.5$
   • B) $2$
   • C) $8$
   • D) It cannot be determined from the given information.
4. The line $y = mx + b$ passes through the points $(-2, -5)$ and $(3, 10)$. What is the value of $m + b$? (Grid-in)
5. The graph of $kx - 3y = 18$ has an $x$-intercept of $6$. What is the value of the constant $k$? (Grid-in)

Level 3: Hard (Advanced SAT)

1. Line $j$ in the $xy$-plane passes through the points $(a, 0)$ and $(0, b)$, where $a$ and $b$ are positive constants. What is the slope of line $j$ in terms of $a$ and $b$?
   • A) $\frac{a}{b}$
   • B) $\frac{b}{a}$
   • C) $-\frac{a}{b}$
   • D) $-\frac{b}{a}$
2. The line $3x - 4y = 24$ forms a right triangle with the $x$-axis and the $y$-axis in the $xy$-plane. What is the area of this triangle? (Grid-in)
3. The function $f$ is defined by $f(x) = 5x - 2$. The graph of $g(x)$ is created by applying the transformation $g(x) = f(x - 3) + 4$. What is the $y$-intercept of $g(x)$? (Grid-in)
4. In the $xy$-plane, line $L_1$ is given by $ax + by = c$, where $a, b,$ and $c$ are strictly positive constants. Which quadrant does line $L_1$ NOT pass through?
   • A) Quadrant I
   • B) Quadrant II
   • C) Quadrant III
   • D) Quadrant IV
5. Line $A$ has the equation $y = 3x - 8$. Line $B$ is perpendicular to Line $A$ and intersects Line $A$ at the point $(4, 4)$. What is the $y$-intercept of Line $B$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. The line $8x + 15y = 120$ intersects the $x$-axis at point $P$ and the $y$-axis at point $Q$. What is the straight-line distance between point $P$ and point $Q$? (Grid-in)
2. Line $M$ has an $x$-intercept of $4$ and a $y$-intercept of $6$. Line $N$ has an $x$-intercept of $6$ and a $y$-intercept of $4$. The two lines intersect at the point $(x, y)$. What is the value of $x + y$? (Grid-in)
3. Two perpendicular lines intersect at the point $(2, 5)$. If the sum of their slopes is $\frac{8}{3}$ and one of the lines has a positive slope, what is the slope of the steeper line?
   • A) $\frac{1}{3}$
   • B) $3$
   • C) $4$
   • D) $5$
4. The line defined by $y = mx + b$ passes through the point $(p, q)$, where $p$ and $q$ are non-zero constants. If the line is shifted to the left by $p$ units and down by $q$ units, the new line passes through the origin. Which of the following must be true?
   • A) $b = 0$
   • B) $m = 1$
   • C) $q = mp + b$
   • D) The original line already passed through the origin.
5. The graph of $y = a|x - 2| + c$ intersects the $x$-axis at exactly two points, $(0, 0)$ and $(4, 0)$. It intersects the $y$-axis at exactly one point. If $a$ and $c$ are constants, what is the value of $\frac{c}{a}$? (Grid-in)

Systems of equations

Systems of linear equations · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. $y = 4x$
   $2x + y = 24$
   The solution to the given system of equations is $(x, y)$. What is the value of $x$? (Grid-in)
2. $x + y = 18$
   $x - y = 6$
   What is the value of $x$?
   • A) $6$
   • B) $12$
   • C) $18$
   • D) $24$
3. $3x - 2y = 10$
   $5x + 2y = 22$
   The solution to the given system is $(x, y)$. What is the value of $8x$? (Grid-in)
4. A system of two linear equations has no solution. Which of the following must be true about the graphs of the two equations in the $xy$-plane?
   • A) They are perpendicular lines.
   • B) They are intersecting lines with different slopes.
   • C) They are parallel lines.
   • D) They are the exact same line.
5. $x = 8$
   $x + 3y = 26$
   The solution to the given system of equations is $(x, y)$. What is the value of $y$? (Grid-in)

Level 2: Medium (Standard SAT)

1. $2x + 3y = 15$
   $x + 2y = 10$
   What is the value of $x + y$?
   • A) $5$
   • B) $10$
   • C) $15$
   • D) $25$
2. $6x + 8y = 10$
   $3x + 4y = c$
   If the given system of equations has infinitely many solutions, what is the value of $c$? (Grid-in)
3. $y = 5x + 4$
   $y = mx - 2$
   If the given system of equations has no solution, what is the value of $m$?
   • A) $-2$
   • B) $4$
   • C) $5$
   • D) $7$
4. $\frac{x}{2} + \frac{y}{3} = 5$
   $x + y = 12$
   The solution to the given system is $(x, y)$. What is the value of $x$? (Grid-in)
5. $4x - y = 9$
   $y = x - 3$
   The solution to the given system is $(x, y)$. What is the value of $y$?
   • A) $-1$
   • B) $1$
   • C) $2$
   • D) $5$

Level 3: Hard (Advanced SAT)

1. $12x + 8y = 40$
   $3x + cy = 10$
   In the given system of equations, $c$ is a constant. If the system has infinitely many solutions, what is the value of $c$? (Grid-in)
2. $kx - 4y = 7$
   $3x + 2y = 5$
   In the given system of equations, $k$ is a constant. If the system has no solution, what is the value of $k$? (Grid-in)
3. $\frac{x}{y} = 5$
   $\frac{x}{ny} = 20$
   The given system of equations relates the positive variables $x$, $y$, and $n$. What is the value of $n$?
   • A) $0.25$
   • B) $4$
   • C) $15$
   • D) $100$
4. $24x + y = 48$
   $6x + y = 72$
   The solution to the given system of equations is $(x, y)$. What is the value of $y$? (Grid-in)
5. $ax + by = 12$
   $2x + 3y = 4$
   The given system of equations has infinitely many solutions, where $a$ and $b$ are constants. What is the value of $a + b$?
   • A) $5$
   • B) $15$
   • C) $20$
   • D) $36$

Level 4: Very Hard (Slightly beyond SAT)

1. $y = 6x + 18$
   $cx - y = -22$
   In the given system of equations, $c$ is a constant. If the system has no solution, what is the value of $c$? (Grid-in)
2. $px + qy = 10$
   $3x + 4y = 8$
   If the system of equations has no solution, which of the following correctly expresses $p$ in terms of $q$?
   • A) $p = \frac{3q}{4}$
   • B) $p = \frac{4q}{3}$
   • C) $p = \frac{10q}{8}$
   • D) $p = \frac{8q}{10}$
3. $48x - 64y = 48 + 24y$
   $rx - 11y = 6$
   In the given system of equations, $r$ is a constant. If the system has infinitely many solutions, what is the value of $r$? (Grid-in)
4. $3x - 5y = 14$
   $ax + by = c$
   The graph of the second equation in the system passes through the points $(0, 4)$ and $(6, 0)$. If the system has exactly one solution, which of the following could be the value of $a$?
   • A) $2$
   • B) $3$
   • C) $-1.8$
   • D) Any real number except $-1.8$
5. At a theater, the price of an adult ticket is $\$10$ and the price of a child ticket is $\$5$. On a certain night, $50$ tickets were sold for a total revenue of $\$350$. How many adult tickets were sold? (Grid-in)

Systems of two linear equations

Systems of linear equations · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. $x = 6$
   $x + 2y = 16$
   The solution to the given system of equations is $(x, y)$. What is the value of $y$? (Grid-in)
2. $y = 3x$
   $2x + y = 20$
   Which ordered pair $(x, y)$ is the solution to the given system of equations?
   • A) $(4, 12)$
   • B) $(5, 15)$
   • C) $(10, 30)$
   • D) $(12, 4)$
3. $x + y = 12$
   $x - y = 4$
   The solution to the given system of equations is $(x, y)$. What is the value of $x$? (Grid-in)
4. $2x + y = 10$
   $4x + 2y = 20$
   How many distinct solutions does the given system of equations have?
   • A) Exactly one
   • B) Exactly two
   • C) Infinitely many
   • D) Zero
5. A family bought 2 adult tickets and 3 child tickets to a fair, paying a total of $\$35$. If an adult ticket costs $\$10$, what is the cost, in dollars, of one child ticket? (Grid-in)

Level 2: Medium (Standard SAT)

1. $24x + y = 48$
   $6x + y = 72$
   The solution to the given system of equations is $(x, y)$. What is the value of $y$? (Grid-in)
2. $y = -3x$
   $4x + y = 15$
   The solution to the given system of equations is $(x, y)$. What is the value of $x$?
   • A) $1$
   • B) $5$
   • C) $15$
   • D) $45$
3. $\frac{x}{2} + \frac{y}{6} = 10$
   $\frac{x}{2} + \frac{y}{3} = 14$
   The solution to the given system of equations is $(x, y)$. What is the value of $y$? (Grid-in)
4. $2x + 3y = 18$
   $2x - y = 6$
   The solution to the given system of equations is $(x, y)$. What is the value of $x + y$?
   • A) $3$
   • B) $4.5$
   • C) $7.5$
   • D) $10.5$
5. $y = 5x + 10$
   One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?
   • A) $5x - y = 12$
   • B) $-5x + y = 10$
   • C) $5x + y = 12$
   • D) $x - 5y = 10$

Level 3: Hard (Advanced SAT)

1. $4x - 9y = 9y + 5$
   $hy = 2 + 4x$
   In the given system of equations, $h$ is a constant. If the system has no solution, what is the value of $h$? (Grid-in)
2. $x + 3y = 10$
   $ax + by = 30$
   In the given system of equations, $a$ and $b$ are constants. If the system has infinitely many solutions, what is the value of $a + b$?
   • A) $4$
   • B) $9$
   • C) $10$
   • D) $12$
3. $3x - 4y = 10$
   $6x - ky = 15$
   In the given system of equations, $k$ is a constant. If the system has no solution, what is the value of $k$? (Grid-in)
4. $ax - 5y = 12$
   $3x - by = 4$
   In the given system of equations, $a$ and $b$ are constants. If the system has infinitely many solutions, what is the value of $a \times b$?
   • A) $9$
   • B) $15$
   • C) $20$
   • D) $27$
5. A theater sold 120 tickets for a play. Adult tickets cost $\$15$ each and child tickets cost $\$8$ each. If the theater collected exactly $\$1,300$ in ticket sales, how many adult tickets were sold? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. $4x + 7y = 10$
   $2x - y = -4$
   The solution to the given system of equations is $(x, y)$. What is the value of $6x + 6y$?
   • A) $-6$
   • B) $0$
   • C) $6$
   • D) $14$
2. $px + qy = 10$
   $2x - 3y = 7$
   In the given system of equations, $p$ and $q$ are constants. If the system has no solution, what is the value of $\frac{p}{q}$? (Grid-in)
3. $x + y = 8$
   $y + z = 12$
   $x + z = 14$
   The solution to the given system of equations is $(x, y, z)$. What is the value of $x + y + z$? (Grid-in)
4. $y = mx + 4$
   $y = 2x + b$
   In the given system of equations, $m$ and $b$ are constants. The system has exactly one solution. Which of the following statements must be true?
   • A) $m = 2$ and $b = 4$
   • B) $m = 2$ and $b \neq 4$
   • C) $m \neq 2$
   • D) $b \neq 4$
5. $\frac{x}{2} + \frac{y}{3} = 5$
   $\frac{x}{4} - \frac{y}{6} = 1$
   The solution to the given system of equations is $(x, y)$. What is the value of $x$? (Grid-in)

Linear inequalities

Linear inequalities · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. $x - 7 \ge 12$. What is the least possible value of $x$? (Grid-in)
2. A delivery truck can carry a maximum load of $2,000$ pounds. If a worker is loading boxes that weigh $50$ pounds each, which inequality represents the number of boxes, $b$, the truck can carry?
   • A) $50b \ge 2000$
   • B) $50b \le 2000$
   • C) $50 + b \le 2000$
   • D) $b \le 50(2000)$
3. $3x + 15 \le 45$. What is the greatest possible value of $x$? (Grid-in)
4. Which of the following points $(x, y)$ is a solution to the inequality $y > x + 2$?
   • A) $(0, 0)$
   • B) $(2, 4)$
   • C) $(3, 6)$
   • D) $(5, 1)$
5. Ty set a goal to walk at least 24 kilometers every day. If Ty walks at an average speed of 4 kilometers per hour, what is the minimum number of hours Ty must walk to fulfill the goal? (Grid-in)

Level 2: Medium (Standard SAT)

1. $-2x + 5 \ge 17$. Which of the following represents all solutions to the given inequality?
   • A) $x \le -6$
   • B) $x \ge -6$
   • C) $x \le 6$
   • D) $x \ge 6$
2. $2(x - 4) < 3x - 10$. What is the least integer value of $x$ that satisfies the inequality? (Grid-in)
3. A student is buying pens and notebooks. Pens cost $\$2$ each and notebooks cost $\$5$ each. The student wants to buy exactly $12$ items and has a maximum budget of $\$35$. Which system of inequalities represents this situation, where $p$ is the number of pens and $n$ is the number of notebooks?
   • A) $p + n \ge 12$ and $2p + 5n \le 35$
   • B) $p + n = 12$ and $2p + 5n \le 35$
   • C) $p + n = 12$ and $2p + 5n \ge 35$
   • D) $p + n \le 12$ and $2p + 5n = 35$
4. Which point is a solution to the system of inequalities $y < 2x - 3$ and $y > -x + 1$?
   • A) $(0, 0)$
   • B) $(1, -1)$
   • C) $(4, 0)$
   • D) $(4, 6)$
5. $y \ge 4x + 8$. For which of the following tables are all the values of $x$ and their corresponding values of $y$ solutions to the given inequality?
   • A) $(2, 19), (4, 30), (6, 41)$
   • B) $(2, 8), (4, 16), (6, 24)$
   • C) $(2, 15), (4, 25), (6, 30)$
   • D) $(2, 16), (4, 24), (6, 32)$

Level 3: Hard (Advanced SAT)

1. $y \le -3x + 15$
   $x \ge 4$
   If $(x, y)$ is a solution to the system of inequalities above, what is the maximum possible value of $y$? (Grid-in)
2. A business owner budgets $\$100$ to purchase supplies. Small boxes cost $\$2$ each, and large boxes cost $\$5$ each. The owner must purchase a minimum of $30$ boxes total. What is the maximum number of large boxes the owner can purchase? (Grid-in)
3. $ax + 5 < 17$. In the given inequality, $a$ is a positive constant. If $x = 3$ is a solution to the inequality, but $x = 4$ is NOT a solution, which of the following could be the value of $a$?
   • A) $2$
   • B) $3.5$
   • C) $4$
   • D) $5$
4. $|x - 2| < 5$. How many integer values of $x$ satisfy the given absolute value inequality? (Grid-in)
5. $y > 13x - 18$. For which of the following tables are all the values of $x$ and their corresponding values of $y$ solutions to the given inequality?
   • A) $(3, 21), (5, 47), (8, 86)$
   • B) $(3, 26), (5, 42), (8, 86)$
   • C) $(3, 16), (5, 42), (8, 81)$
   • D) $(3, 26), (5, 52), (8, 91)$

Level 4: Very Hard (Slightly beyond SAT)

1. $y \le 5x + c$. In the given inequality, $c$ is a constant. If the point $(3, 10)$ is a solution to the inequality, what is the minimum possible value of $c$? (Grid-in)
2. $\frac{x}{3} - \frac{x}{4} > 1$. What is the least integer value of $x$ that satisfies the inequality? (Grid-in)
3. $y \ge \frac{1}{2}x + 3$
   $y \le \frac{1}{2}x - 2$
   How many solutions $(x, y)$ exist for the given system of inequalities in the $xy$-plane?
   • A) Zero
   • B) Exactly one
   • C) Exactly two
   • D) Infinitely many
4. $y \le -x + 4$
   $y \ge x$
   $y \ge 0$
   The solution to the given system of inequalities forms a triangular region in the $xy$-plane. What is the area of this region? (Grid-in)
5. $y \le -2x + 20$
   $y \ge 3x - 15$
   If $(x, y)$ is a solution to the system of inequalities above, what is the maximum possible value of $x$? (Grid-in)

Linear equations, systems and modelling

Stretch sheets · 10 questions · 10 SAT-style algebra questions with an HL twist — parameter matching, invertibility conditions, and linear systems.

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational)

1. A line has equation $2x - 3y = 12$. What is its $y$-intercept?
   A) $-4$
   B) $4$
   C) $-6$
   D) $6$
2. If $f(x)=5x-7$, what is the value of $x$ for which $f(x)=13$?
   A) $3$
   B) $4$
   C) $-4$
   D) $20$
3. A taxi charges a fixed $\$3$ pickup fee plus $\$1.50$ per kilometre. Determine the exact cost, in dollars, of an $8$ km trip. (SPR)

Level 2: Medium (Standard)

1. Solve the system $3x + 2y = 12$, $x - y = 1$. Which is the value of $y$?
   A) $\tfrac{9}{5}$
   B) $\tfrac{7}{5}$
   C) $2$
   D) $3$
2. For what value of $k$ does the system $x + ky = 3$, $2x + 4y = 6$ have infinitely many solutions?
   A) $1$
   B) $2$
   C) $3$
   D) $4$
3. The line $y = 2x + b$ passes through $(3, 11)$. Determine the exact value of $b$. (SPR)
4. A rope $L$ metres long is cut into two pieces so the longer piece is $2$ m more than twice the shorter. If the shorter piece is $s$ m, which equation is correct?
   A) $L = 3s + 2$
   B) $L = 3s - 2$
   C) $L = 2s + 2$
   D) $L = s + 2$

Level 3: Hard (Advanced)

1. The system $\begin{cases}3x - 2y = 8 \\ 6x + ky = 5\end{cases}$ has no solution. Determine the value of $k$.
   A) $-4$
   B) $4$
   C) $-8$
   D) $8$
2. Line $\ell$ has slope $m$ and passes through $(2,5)$ and $(6,-3)$. Determine the exact value of $m$. (SPR)
3. A savings account starts at $\$400$ and grows by a fixed $\$25$ each month. After how many months $t$ does the balance first exceed $\$1000$?
   A) $23$
   B) $24$
   C) $25$
   D) $26$

Linear equations, systems and real-world models

Stretch sheets · 10 questions · 10 SAT-style algebra questions themed around HL AI's applied modelling: cost-revenue, break-even points, and mixture problems.

Practice Questions

Level 1: Easy (Foundational)

1. Solve $4x - 5 = 11$.
   A) $3$
   B) $4$
   C) $5$
   D) $16$
2. Which pair $(x,y)$ satisfies $x + y = 10$ and $x - y = 2$?
   A) $(6, 4)$
   B) $(4, 6)$
   C) $(5, 5)$
   D) $(7, 3)$
3. A car rental is $\$40$ per day plus $\$0.20$ per km. Determine the exact cost, in dollars, of a $3$-day rental with $250$ km. (SPR)

Level 2: Medium (Standard)

1. A company's cost is $C = 20n + 500$ and revenue is $R = 45n$. Determine the exact break-even $n$.
   A) $18$
   B) $20$
   C) $25$
   D) $30$
2. A $30\%$ acid solution is mixed with a $60\%$ acid solution to form $10$ L of $45\%$ acid. How many litres of the $30\%$ solution are needed?
   A) $3$ L
   B) $4$ L
   C) $5$ L
   D) $6$ L
3. A phone plan is $\$20$/mo plus $\$0.10$ per minute. Determine the exact minutes for a bill of $\$45$. (SPR)
4. Line $\ell$ passes through $(0, -3)$ and $(4, 5)$. Which is the equation of $\ell$?
   A) $y = 2x - 3$
   B) $y = 2x + 3$
   C) $y = -2x - 3$
   D) $y = \tfrac{1}{2}x - 3$

Level 3: Hard (Advanced)

1. Two runners start together; one runs at $6$ km/h and the other at $8$ km/h in the same direction. After how many hours is the faster one $5$ km ahead?
   A) $2.0$
   B) $2.5$
   C) $3.0$
   D) $5.0$
2. A budget line is $2x + 3y = 60$. Determine the maximum integer value of $y$ when $x \ge 3$. (SPR)
3. Three consecutive integers sum to $126$. Determine the largest integer.
   A) $41$
   B) $42$
   C) $43$
   D) $44$